Chứng minh $1+\dfrac{1}{2}+\dfrac{1}{3}+\dots+\dfrac{1}{32} >3.$

Đáp án:

 `A > 3`

Giải thích các bước giải:

 Gọi `A = 1 + 1/2 + 1/3 + .... + 1/32`

          ` = 1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9+ 1/10 + ... + 1/16) + (1/17 + 1/18 + 1/19 + .... + 1/32)`

Xét tổng :

* `1/2 + 1/3 + 1/4 > 1/2 + 1/4 + 1/4`

`=> 1/2 + 1/3 + 1/4 > 1`

* `1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8`

`=> 1/5 + 1/6 + 1/7 + 1/8 > 1/2`

* `1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 > 1/16 + 1/16 + 1/16 + .... + 1/16`

`=> 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 >1/2`

* `1/17 + 1/18 + 1/19 + .... + 1/32 > 1/32 + 1/32 + 1/32 + .... + 1/32`

`=> 1/17 + 1/18 + 1/19 + .... + 1/32 > 1/2`

`=> A > 1 + 1 + 1/2 + 1/2 + 1/2`

`=> A > 2 + 3/2 > 2 + 1`

`=> A > 3` (đpcm)